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 Mat. Sb., 1994, Volume 185, Number 8, Pages 81–102 (Mi msb919)

This article is cited in 28 scientific papers (total in 28 papers)

On approximation by families of linear polynomial operators in $l_P$-spaces, $0<p<1$

K. V. Runovskii

T. H. Shevchenko Chernihiv State Pedagogical Institute

Abstract: New applications are given for a method of approximation by families of linear polynomial operators in $L_p$-spaces with $0<p<1$ that was developed by the author in previous papers. In particular, a direct theorem and an inverse theorem on approximation 'by angle' of trigonometric polynomials are proved in the case of higher-order moduli of smoothness.

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English version:
Russian Academy of Sciences. Sbornik. Mathematics, 1995, 82:2, 441–459

Bibliographic databases:

UDC: 517.51
MSC: Primary 41A10, 41A27, 41A63, 42A10; Secondary 41A50, 42B99
Received: 27.09.1993

Citation: K. V. Runovskii, “On approximation by families of linear polynomial operators in $l_P$-spaces, $0<p<1$”, Mat. Sb., 185:8 (1994), 81–102; Russian Acad. Sci. Sb. Math., 82:2 (1995), 441–459

Citation in format AMSBIB
\Bibitem{Run94} \by K.~V.~Runovskii \paper On approximation by families of linear polynomial operators in $l_P$-spaces, $0<p<1$ \jour Mat. Sb. \yr 1994 \vol 185 \issue 8 \pages 81--102 \mathnet{http://mi.mathnet.ru/msb919} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1302624} \zmath{https://zbmath.org/?q=an:0841.42001} \transl \jour Russian Acad. Sci. Sb. Math. \yr 1995 \vol 82 \issue 2 \pages 441--459 \crossref{https://doi.org/10.1070/SM1995v082n02ABEH003574} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1995RV83000012} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. N. N. Pustovoitov, “Estimates of the best approximations of periodic functions by trigonometric polynomials in terms of averaged differences and the multidimensional Jackson's theorem”, Sb. Math., 188:10 (1997), 1507–1520
2. Burinska Z., Runovski K., Schmeisser H., “On the Method of Approximation by Families of Linear Polynomial Operators”, Z. Anal. ihre. Anwend., 19:3 (2000), 677–693
3. Ditzian Z., Runovskii K., “Realization and Smoothness Related to the Laplacian”, Acta Math. Hung., 93:3 (2001), 189–223
4. S. B. Vakarchuk, “Exact Constants in Jackson-type Inequalities and Exact Values of Widths”, Math. Notes, 78:5 (2005), 735–739
5. Runovski K., Rystsov I., Schmeisser H.-J., “Computational Aspects of a Method of Stochastic Approximation”, Z. Anal. ihre. Anwend., 25:3 (2006), 367–383
6. Konstantin Runovski, Hans-Jürgen Schmeisser, “On Approximation Methods Generated by Bochner-Riesz Kernels”, J Fourier Anal Appl, 14:1 (2008), 16
7. K. Runovski, H.-J. Schmeisser, “On convergence of families of linear polynomial operators generated by matrices of multipliers”, Eurasian Math. J., 1:3 (2010), 112–133
8. V. I. Ivanov, “Direct and inverse theorems in approximation theory for periodic functions in S. B. Stechkins papers and the development of these theorems”, Proc. Steklov Inst. Math. (Suppl.), 273, suppl. 1 (2011), S1–S13
9. K.  Runovski, H.-J. Schmeisser, “On families of linear polynomial operators generated by Riesz kernels”, Eurasian Math. J., 1:4 (2010), 124–139
10. F. Dai, Z. Ditzian, “Jackson theorem in , for functions on the sphere”, Journal of Approximation Theory, 162:2 (2010), 382
11. Vladimir Rukasov, Konstantin Runovski, Hans-Jürgen Schmeisser, “Approximation by families of linear trigonometric polynomial operators and smoothness properties of functions”, Math. Nachr, 2011, n/a
12. M. Sh. Shabozov, G. A. Yusupov, “Exact constants in Jackson-type inequalities and exact values of the widths of some classes of functions in $L_2$”, Siberian Math. J., 52:6 (2011), 1124–1136
13. K. Runovski, H.-J. Schmeisser, “Methods of trigonometric approximation and generalized smoothness. I”, Eurasian Math. J., 2:3 (2011), 98–124
14. Vakarchuk S.B. Doronin V.G., “Best Mean Square Approximations by Entire Functions of Finite Degree on a Straight Line and Exact Values of Mean Widths of Functional Classes”, Ukr. Math. J., 62:8 (2011), 1199–1212
15. S. B. Vakarchuk, V. I. Zabutnaya, “Jackson–Stechkin Type Inequalities for Special Moduli of Continuity and Widths of Function Classes in the Space $L_2$”, Math. Notes, 92:4 (2012), 458–472
16. Akgun R., “Approximation by Polynomials in Rearrangement Invariant Quasi Banach Function Spaces”, Banach J. Math. Anal., 6:2 (2012), 113–131
17. Vakarchuk S.B. Zabutnaya V.I., “On the Best Polynomial Approximation in the Space l (2) and Widths of Some Classes of Functions”, Ukr. Math. J., 64:8 (2013), 1168–1176
18. G. A. Yusupov, “Best polynomial approximations and widths of certain classes of functions in the space $L_2$”, Eurasian Math. J., 4:3 (2013), 120–126
19. M. R. Langarshoev, “Tochnye neravenstva tipa Dzheksona–Stechkina i znacheniya poperechnikov nekotorykh klassov funktsii v prostranstve $L_{2}$”, Model. i analiz inform. sistem, 20:5 (2013), 90–105
20. Shabozov M.Sh., Vakarchuk S.B., Zabutnaya V.I., “Sharp Jackson-Stechkin Type Inequalities for Periodic Functions in l-2 and Widths of Function Classes”, Dokl. Math., 88:1 (2013), 478–481
21. K. Tukhliev, “O priblizhenii periodicheskikh funktsii v $L_2$ i znacheniyakh poperechnikov nekotorykh klassov funktsii”, Model. i analiz inform. sistem, 22:1 (2015), 127–143
22. S. B. Vakarchuk, “Generalized Smoothness Characteristics in Jackson-Type Inequalities and Widths of Classes of Functions in $L_2$”, Math. Notes, 98:4 (2015), 572–588
23. S. B. Vakarchuk, V. I. Zabutnaya, “Inequalities between Best Polynomial Approximations and Some Smoothness Characteristics in the Space $L_2$ and Widths of Classes of Functions”, Math. Notes, 99:2 (2016), 222–242
24. K. V. Runovskii, N. V. Omel'chenko, “Mixed Generalized Modulus of Smoothness and Approximation by the “Angle” of Trigonometric Polynomials”, Math. Notes, 100:3 (2016), 448–457
25. K. V. Runovski, “Trigonometric polynomial approximation, $K$-functionals and generalized moduli of smoothness”, Sb. Math., 208:2 (2017), 237–254
26. Vakarchuk S.B., “Jackson-Type Inequalities With Generalized Modulus of Continuity and Exact Values of the N-Widths For the Classes of (Psi, Beta)-Differentiable Functions in l-2. III”, Ukr. Math. J., 68:10 (2017), 1495–1518
27. Kolomoitsev Yu., “on Moduli of Smoothness and Averaged Differences of Fractional Order”, Fract. Calc. Appl. Anal., 20:4 (2017), 988–1009
28. M. Sh. Shabozov, A. A. Shabozova, “Nekotorye tochnye neravenstva tipa Dzheksona - Stechkina dlya periodicheskikh differentsiruemykh v smysle Veilya funktsii v $L_2$”, Tr. IMM UrO RAN, 25, no. 4, 2019, 255–264
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